A172099 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = x^(2*n-1)*p(n-1, x) + p(n-2, x) with p(0, x) = 1 and p(1, x) = 1 + x, read by rows.

1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1

Offset: 0

Author

Roger L. Bagula, Jan 25 2010

Keywords

• nonn
• tabf

Comments

There are n^2 + 1 terms in row n, for n >= 0. - G. C. Greubel, Apr 07 2022

Examples

			Irregular triangle begins as:
  1;
  1, 1;
  1, 0, 0, 1, 1;
  1, 1, 0, 0, 0, 1, 0, 0, 1, 1;
  1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1;
  1, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1;
		

Links

• G. C. Greubel, Rows n = 0..30 of the irregular triangle, flattened

Crossrefs

Cf. A000045, A002522, A172098.

Programs

Mathematica

p[n_, x_]:= p[n, x]= If[n<2, n*x+1, x^(2*n-1)*p[n-1, x] + p[n-2, x]];
Table[CoefficientList[p[n, x], x], {n,0,10}]//Flatten

SageMath

@CachedFunction
def p(n,x):
    if (n<2): return n*x+1
    else: return x^(2*n-1)*p(n-1, x) + p(n-2, x)
def T(n,k): return ( p(n,x) ).series(x, n^2+2).list()[k]
flatten([[T(n,k) for k in (0..n^2)] for n in (0..12)]) # G. C. Greubel, Apr 07 2022

Formula

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = x^(2*n-1)*p(n-1, x) + p(n-2, x) with p(0, x) = 1 and p(1, x) = 1 + x.

Sum_{k=0..n^2} T(n, k) = Fibonacci(n+2) (row sums). - G. C. Greubel, Apr 07 2022

Extensions

Edited by G. C. Greubel, Apr 07 2022